[Paper Review] Strong convergence rate in averaging principle for stochastic hyperbolic-parabolic equations with two time-scales

Hongbo Fu, Li Wan+2 more|arXiv (Cornell University)|Nov 28, 2016

Stability and Controllability of Differential Equations19 references3 citations

TL;DR

This paper establishes a strong convergence rate of order $1/2$ for the averaging principle in stochastic hyperbolic-parabolic equations with two time-scales, where both slow and fast components are driven by multiplicative noise. By analyzing the invariant measure and ergodicity of the fast component, the authors derive a simplified averaged equation and prove optimal convergence via energy estimates and Gronwall-type inequalities, significantly improving upon the prior $1/4$ rate.

ABSTRACT

In this article, we investigate averaging principle for stochastic hyperbolic-parabolic equations with two time-scales, in which both the slow and fast components are perturbed by multiplicative noises. Particularly, we prove that the rate of strong convergence for the slow component to the averaged dynamics is of order $1/2$, which significantly improves the order $1/4$ established in our previous work.

Motivation & Objective

• To establish the strong convergence rate of the slow component to its averaged dynamics in a two-time-scale stochastic hyperbolic-parabolic system.
• To address the gap in convergence rates for SPDEs with multiplicative noise, particularly in infinite-dimensional settings.
• To improve upon the prior $1/4$ convergence rate by proving an optimal $1/2$ rate under suitable ergodicity and regularity conditions.
• To analyze the interplay between multiplicative noise in both slow and fast components and its impact on averaging dynamics.

Proposed method

• Formulate a coupled system of stochastic hyperbolic and parabolic equations with two time-scales, driven by multiplicative noise.
• Use the averaging principle to derive a simplified effective equation for the slow component by integrating out the fast component.
• Establish the existence and uniqueness of invariant measures and ergodicity for the fast component under suitable conditions on the drift and diffusion coefficients.
• Apply energy estimates and Gronwall-type inequalities to control the difference between the original and averaged systems.
• Derive bounds on the derivative of the averaged drift via perturbation analysis of the fast component's response to slow variable variations.
• Leverage exponential decay estimates and stochastic calculus to prove the $1/2$ convergence rate in the strong sense.

Experimental results

Research questions

• RQ1 What is the optimal strong convergence rate for the slow component in a two-time-scale stochastic hyperbolic-parabolic system with multiplicative noise?
• RQ2 Can the convergence rate be improved from $1/4$ to $1/2$ when both slow and fast components are perturbed by multiplicative noise?
• RQ3 How does the ergodicity and invariant measure of the fast component influence the averaging dynamics of the slow component?
• RQ4 What role does the multiplicative noise structure play in determining the convergence rate of the averaging principle?
• RQ5 Can the convergence rate be established in infinite-dimensional SPDEs with non-additive noise, and if so, under what conditions?

Key findings

• The strong convergence rate of the slow component to the averaged dynamics is proven to be of order $1/2$, which is optimal and significantly improves the prior $1/4$ rate.
• The convergence rate is achieved despite both slow and fast components being driven by multiplicative noise, a setting where rates typically degrade.
• The proof relies on establishing exponential decay in the derivative of the averaged drift with respect to the slow variable, uniformly in the fast component.
• The authors derive a uniform bound on the difference between the averaged drift and its derivative, showing $\|\tilde{F}_{t_0}(x_1,y,t) - \tilde{F}_{t_0}(x_2,y,t)\| \leq C(1+\|y\|)\|x_1 - x_2\|e^{-ct}$.
• The convergence result holds under conditions ensuring the fast component's ergodicity and the existence of a unique invariant measure.
• The analysis confirms that the $1/2$ rate is attainable in infinite-dimensional SPDEs, answering an open question in the field.